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A generalized least squares fast transversal filter algorithm for the decision feedback equalization of dispersive channels
Signal Processing 21 (1990) 241-250 Elsevier
241
A G E N E R A L I Z E D LEAST S Q U A R E S FAST TRANSVERSAL FILTER A L G O R I T H M FOR THE D E C I S I O N FEEDBACK E Q U A L I Z A T I O N OF DISPERSIVE C H A N N E L S D.K. M E H R A Department of Electronics and Computer Engineering, University of Roorkee, Roorkee, India 247-667 Received 29 February 1988 Revised 27 December 1988 and 25 May 1989
Abstract. Recently computationally efficient recursive least squares adaptive filtering algorithms have been introduced in transversal filter form as well as in lattice form. Fast transversal filter implementation of the generalized recursive least squares adaptive algorithm, which is appropriate for multichannel adaptive filtering and estimation, is presented in this paper. The above algorithm is derived in particular for the decision feedback equalization of dispersive channels, when the number of stages in the forward and backward filters are different. The algorithm considered in this paper is advantageous over the generalized least squares lattice algorithm considered by Ling and Proakis, because of the less computational effort required. The convergence rate of~the mean square error for the decision feedback equalization of QPSK data transmitted through the dispersive channel using the generalized fast transversal filter algorithm is obtained using computer simulations. Zusammenfassung. Kfirzlich sind recheneffiziente rekursive adaptive Filteralgorithmen auf der Basis Kleinster Quadrate sowohl in Transversal als auch in lattice-Form eingeffihrt worden. In diesem Beitrag wird eine schnelle Realisierung als Transversalfilter ffir den verallgemeinerten rekursiven adaptiven Kleinste-Quadrate-Algorithmus vorgestellt, der ffir die mehrkanalige adaptive Filterung und Sch~itzung geeignet ist. Der genannte Algorithmus wird insbesondere hergeleitet ffir die Entzerrung dispersiver Kanfile mit Entscheidungs-Rfickkopplung ffir den Fall, da D die Anzahlen der Stufen in den Vorwiirts- und Rfickwartsfiltern unterschiedlich sind. Der hier behandelte Algorithmus weist gegenfiber dem entsprechenden Latticealgorithmus, wie ihn Ling und Proakis betrachten, Vorteile auf, da er weniger Rechenaufwand erfordert. Aus Rechnersimulationen werden Angaben fiber die Konvergenzgeschwindigkeit des mittleren quadratischen Fehlers des schnellen verallgemeinerten Transversalfilteralgorithmus bei einer QSPK-Daten-Entzerrung mit Entscheidungsrfickkopplung und nach der l]bertragung fiber einen dispersiven kanal gewonnen.
R6sum6. R6cemment des algorithmes r6cursifs de filtrage adaptatif aux moindres carr6s efficaces au niveau calcul ont 6t6 introduits aussi bien sous la forme transversale qu'en treillis. L'implantation sur un filtre transversal rapide de l'algorithme r6cursif adaptatif aux moindres carr6s g6n6ralis6s, qui se pr&e bien aux op6rations de filtrage adaptatif multicanal et d'estimation est pr6sent6 dans cet article. Cet algorithme est sp6cifi6 en particulier pour l'egalisation par d6cision en mode r6trograde de cannaux dispersifs, quand les tailles des filtres avant et arri~re sont diff6rentes. L'algorithme consid6r6 dans cet article est plus avantageux que l'algorithme aux moindres carr6s g6n6ralis6s en treillis pr6sent6 par Ling et Proakis, car il n6cessite moins de calculs. Le taux de convergence de l'erreur quadratique moyenne lors de 1'6galisation en mode de d6cision r6trograde de donn6es QPSK transmises fi travers un canal dispersif ~ I'aide de I'algorithme de filtre transversal rapide g6n6ralis6 est obtenue a l'aide de simulations sur ordinateur.
Keywords. Recursive estimation, generalized fast transversal filtering, decision feedback equalization.
1. Introduction In high speed
digital communication,
use of the available
channel
bandwidth
limited by the presence of the intersymbol
efficient is o f t e n interfer-
0165-1684/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
ence caused by the dispersion of the channel. The linear equalizer [8] widely used for the equalization of such channels employs a transversal filter structure, whose tap coefficients are adjusted adaptively using the gradient type LMS algorithm of
242
D.K. Mehra / A generalized.fast transversal filter algorithm
Widrow and Hoff [9]. On channels that have spectral nulls, the linear equalizer yields poor error rate performance. In such situations, the decision feedback equalizer which has a very similar computational complexity, but a better error rate performance, is used. The simplest adaptive DFE consists of two transversal filters, a feedforward filter and a feedback filter, and uses the WidrowHoff type LMS algorithm to adapt its coefficients [9]. Although the computational complexity of these algorithms is of the order of 2 N to 3N, where N is the number of equalizer tap coefficients, these algorithms are slower to converge initially. Initial convergence of the algorithm based on the Kalman algorithm [3] is very fast, but a disadvantage of the Kalman algorithm is its computational complexity, where N 2 operations are to be performed per iteration. Recently computationally efficient recursive least square (RLS) adaptive algorithms have been introduced. These algorithms require a number of arithmetic operations per iteration that is proportional to the number of variable parameters in the adaptive filter. These algorithms simultaneously offer the improved convergence properties over LMS algorithms. RLS algorithms have been introduced in transversal filter form as well as in lattice filter form. Recently least squares lattice DFE and gradient lattice DFE algorithms have been developed and their application to the equalization of fixed and time varying multipath channels have also been considered [6, 7]. Lattice algorithms implement RLS algorithms by simultaneous order and time updating. In equalizer applications, lower order solutions are generally not essential, a reduction in computational r'equirement in comparison to the lattice form is achieved through the use of transversal filter algorithms. Fast transversal filter algorithms (FTF) introduced in [1] are the fastest known RLS algorithms and when considered for linear equalization, their computational complexity is proportional to 7N. In this paper, we have developed an FTF DFE algorithm, which differs from the conventional multichannel FTF algorithm [ 1] which is restricted Signa~ Processing
to have the same number of stages in the feedforward and the feedback stages. The FTF algorithm derived is applicable in a general situation, when the number of stages in forward and backward filters are different. We also consider the application of the above algorithm to the DFE of fixed dispersive channel, when QPSK data is transmitted over such channels. Convergence rate of the mean squares error for the above situation is determined using computer simulations. The organisation of this paper is as follows. The generalized least squares filtering problem is introduced in Section 2 and the generalized FTF algorithm is derived in Section 3. The computational complexity of the algorithm when applied to the decision feedback equalization of dispersive channels and its comparison to some other DFE algorithms is given in Section 4. In Section 5 we finally consider the application of the above algorithm and also give some simulation results.
2. Generalized least squares filtering The classical least squares problem is to find the adaptive filter coefficients at each time instant so as to minimize the accumulation of squared error between the filter output and the desired output up to that time instant. For example in an equalizer adjustment algorithm, suppose u(0), u ( 1 ) , . . . , u(n) is a training sequence of complex data symbols transmitted over a channel represented by a tapped delay line filter structure with tap spacing equal to the symbol interval. Channel introduces intersymbol interference and additive noise. Further assume that the sequence {u(n)} transmitted over a channel results in the received sequence as {y(n)}. To retrieve the transmitted sequence, the DFE uses the linear combination of the received signal y(n) and its past values y(n - i), i = 1 , 2 , . . . , N~ - 1. In addition, it uses the information about previously decoded values d ( n - 1), d(n - 2 ) , . . . , d(n - N2) of the transmitted sequence.
D.K. Mehra / A generalizedfast transversalfilter algorithm We thus define the following m-dimensional vector Xm(n), consisting of time delayed samples of y(n) and the past decision as
243
where R,,,(n) is an m x m correlation matrix given by
R,n(n)= ~ A n iXm(j)X~(j) , i
y(n)
(5)
0
and the cross correlation vector ~,,(n) is given by
y(n-1)
n
X~(n) = v ( n - N , + I d(n-1) d(n - 2 )
,
(1)
(6)
X.,(n) defined in (1) is an (N~+ N2)-dimensional vector and, similar to the ease of
The data vector
d(n - N2) where
m=NI+N2. The least squares problem in DFE is then to find the m-dimensional vector Cm(n) of equalizer tap values, which minimizes the following sum of squared errors:
A" i e,,(j) e*(j),
~ , , ( n ) = ~" An i[d*(j)Xm(j)].
(2)
the linear equalizer, a certain shifting property exists for X,.(n). Without loss of generality we assume that N~ ~> N2, and using the permutation matrices SM and QM defined in [12] for deriving the fast Kalman algorithm, we ean show, using the definition of the extended vector
~'m(n + 1) = [y(n + 1 ) , y ( n ) , . . . , y ( n -- N, + 1), d ( n ) , . . , d ( n - N:)] x,
(7)
that
.j-o
SMRM(n + 1) = L X.,(n where the exponential forgetting factor A is such that 0~
(3)
and * denotes the complex conjugate operation. In (3)
where M = m + 2 , p = 2 and ep(n + l) is the vector consisting o f p new elements entering the equalizer given by
Similarly
-Xm(n+l)] QMXM(n + 1) . . . . ,
(9)
p~(n + l) J
C,,(n) = [ c , ( n ) , . . . , CN,(n), CN,+,(n),
where pp(n + 1) is the vector consisting o f p deleted old elements given by
. . . . cN,+N~(n)] ~, where c~(n), i= 1 , 2 , . . . , Nj, are the tap gains of the forward filter, and cN,+~(n), i = 1, 2 , . . . , N2, are the tap gains of the backward filter. Using the definition of the complex gradient operator given in Appendix A, we can show that the optimum value of the vector Cm(n) which minimizes the value of (2) is given by
C,,(n) = -Rm'(n)~m(n),
(8)
[v(n+l)] e p ( n + l ) = [ _ " d(n) J"
the a posteriori error residual, e,,(n), is
era(n) = d(n)+ C~(n)Xm(n)
'
(4)
[Y(n-N,+I) 1 d ( n - N 2 ) 3"
Pr(n+l)=[_
3. Generalized FFF algorithm
It is convenient to carry out the derivation of the algorithm in terms of the a priori error residual Vol 21. No 3, No~ember 1990
244
D.K. Mehra / A generalizedfast transversalfilter algorithm
e,. (n) defined as
e,.(n)=d(n)+C~(n-1)X.,(n).
(10)
In addition, a priori and a posteriori forward prediction error residual vectors are defined as
er~(n)=ep(n)+A~p(n-1)Xm(n-1),
(lla)
e~(n)=er(n)+a~p(n)X~.(n-1),
(llb)
where the m xp tap gain matrix Amp(n) of the foreword prediction error filter is recursively adapted to minimize the trace of
It is shown in Appendix B that the tap gains of the forward, backward and equalizer filter can be recursively updated as
A.,p(n+l)=a,.p(n)+ Wm(n)[e~(n+l)] H, Bmp(n + 1) = Br.p(n)+ Wm(n + 1)[eb(n + 1)] H,
(20) C,.(n+l)=Cm(n)+Wm(n+l)e*(n+l),
j=o
n-j
f
•
f
•
H
em(J)(em(J) ) .
Rm(n-1)Amp(n)= -Vmp(n), r
Wm(n + 1) = - 1 R ~ ' ( n ) X m ( n + l ) .
(12)
It is shown in Appendix B that the optimum value of Amp(n) is given by
(22)
Similarly we may define the extended gain vector as 1
(13)
where the cross-correlation matrix 7~p(n) is given by
(21)
where the gain vector Wm(n + 1) of the algorithm is defined as
n
2 A
(19)
l~'M(n + 1) = --~-/~ml(n).~M(n + 1),
(23)
where/~M (n) is the correlation matrix of the extended vector -~M(n + 1) given by
n
V~p(n)= ~ 3." Jxm(j--1)eHp(j).
(14)
j = 0
It can easily be shown that the minimum value of (12) can be recursively updated as
E~(n + 1) = AE~(n) + eCm(n+ 1)[e~(.+,)] H. (15)
/~M(n+l)= ~ An-J[f(M(n+I)][XM(n+I)]H. j-o In Appendix B, it is shown that the extended gain vector 1~,"M(n + 1) can be recursively updated as ~.(n+l)
_(I/A)[Ef.,(n)]_ '
The a priori and a posteriori backward error residuals are defined as
eb(n)=pp(n)+B~p(n-l)Xm(n),
(16a)
eb(n) = pp(n) + B~p(n)X,.(n),
(16b)
where the m x p tap gain matrix Bmp(n ) of the backward prediction error filter is recursively updated to minimize the trace of A n--Jeb (j)[ eb.n(j) ]H.
= ST~
.
.
.
er.,(n + 1)] .
.
.
(24)
or QMWM(n+l)
[
Wm(n+l)-(1/A)B,,w(n)[E~(n)]-'
(17)
...
Eb(n + 1) = AEb.,(n) + eb(n + 1)[eb(n + 1)] H. (18) Signal
Processing
eb(n+ 1)] ...
- (1/A)[E~(n)]-t
.
eb(n + 1)
j--0 It may be shown easily that the minimum value of (17) can be recursively updated as
.
W,,,(n)_(1/A)Amp(n)[Ef(n)] t er,,,(n+l)
(25) Define
QM[ffCM(n+ I)] . . . .
.
(26)
D.K. Mehra / A generalized fast transversal filter algorithm
5. Simulation results
From (25) and (26) we get W~,(n + 1) = ff'm(n+l)-Bmp(n)l.%(n+l), (27)
eb(n + 1) = -AEb(n)/xp(n + 1).
(28)
Further, the scalar constant am(n) is defined as
am(n) = 1 - X~(n) W,~(n)
245
The generalized FTF algorithm considered above is used for the adaptive equalization of two linear dispersive communication channels, whose discrete time equivalent channel impulse responses are
(29)
Channel 1
(0.30, 0.90, 0.30),
and it is shown in Appendix B that ~,.(n) can be recursively updated as
Channel 2
(0.408, 0.815, 0.408).
ct,,(n+ l)=ctm(n) + 1 [e f ( n + 1)]H[Ef (n)]_~ef ( n+ 1) + [eb~(n + 1)]"tZp(n + 1).
(30)
The a posteriori error residual vectors can be updated using the a priori error residual vector and scalar constant a~(n) as
e~(n + 1) - er''(n + 1) ~m(n)
e b ( n +1)
(31) '
eb(n + 1) a.,(n + 1)'
Channel 1 has eigenvalue ratio of 25, while Channel 2 has eigen value ratio of a. The input to the channel is a uniformly distributed complex sequence, where each element of the sequence takes any one of the four values from the set (l+j,-l+j,-1-j,l-j) thus representing the QPSK data. For the two channels under consideration, complex additive Gaussian noise with variance 2 =0.001 or 0.01 corrupts the signal. The decision feedback equalizer is assumed to have 9 forward taps and 2 backward taps. Initial conditions for the generalized FTF algorithm are
(32) 0
e . , ( n + 1)
em(n+ l)=ol,.(n+ l ).
(33)
Wm(O ) = Cm(O )
=Omxl,
A.,p(O) = Brae(O) = Ore×p, 4. Generalized FFF algorithm and its computational complexity We can rewrite the generalized FTF algorithm using the scalar constant %,(n) defined as 1 w(n)
=
~(n)"
The algorithm is listed in Table 1, along with the computational complexity of each step, when A = 1. The computational complexity has been computed in terms of required number of multiplications. The complexity of some of the adaptive DFE algorithms is listed in Table 2.
where 0 is a null matrix of dimensions as indicated. The mean squares error (MSE) convergence rate performance for the generalized FTF and LMS algorithms is obtained by averaging the MSE over 100 individual learning curves. The LMS algorithm uses a constant step size ~ = 0.025. Figures 1 and 2 show the convergence rate of the generalized FTF algorithm with A =1 and of the LMS algorithm for Channels 1 and 2, respectively. It may be seen from the figures that for the G F T F algorithm, the convergence rate is independent of the channel eigenvalue ratio and the G F T F algorithm converges significantly faster than the LMS algorithm. Vol. 21, No. 3, November |990
246
D.K. Mehra
A generalized fast transversal fiher algorithm
Table 1
Generalized FTF algorithm and c o m p u t a t i o n a l complexity Computation
Complexity
ef,,,(n+ 1) = el,(n+ 1)+ A~w(n)X,,,(n)
mp
Ee,.(n + 1) = e~,,(n + 1) y,,,(n)
P
-(lla)[U,.(n)]' l ~ a (n +
l)
= sT
ef'"(n+l)]...
mp + p2
...
W.,(n)-(lt.t)A,..(n)[Ef.,(n)]
'
e~,,(n + 1)
A,,,~,(n+ 1) = A,,,p(n) + W,,,(n)[e~,,(n+ 1)] H
mp
Ef.,(n + 1) = AEf.,(n)+ e~,,(n + 1)[e~,,(n + 1)] H
p:
[ #,°(n+,)] O~[#M(n + 1)] .
. . .
/z¢,(n + 1) _] W,,,(n + 1) = l,V,,,(n+l)-B.,p(n)~p(n+l)
mp p2
e~,(n + 1 ) = -AE~,(n)~.(n + 1) v,,,(n)
~M (n + 1) - 1 + (1/A)[ ef.,(n + 1)]H[ Elm(n)] - ' er,,,(n + 1) y,.(n + 1)
yM(n+ l) 1 + [eb,(n + l)]H/z/,(n + 1)VM (n + 1)
eb,,,(n + 1) = e~,(n + 1)ym(n + 1)
P 2p
E~,(n + 1) = A E,b,,(n ) + e,b,,(n + 1)[e~,( n + 1)]H
P pZ
B,,,(n + l) = B.,(n)+ W,,,(n + l)[~b,(n + l)] H
mp
e,,,(n + l) = d(n + 1)+ C~,(n)X,.,(n + I)
m
e,.(n+ l ) = e , . ( n + l)y.,(n+ l)
1
C,,(n
+ 1) =
C,,,(n)+ W,,,(n + l)e*,(n + 1)
m
6. Conclusion
Table 2 C o m p u t a t i o n a l complexity of adaptive D F E algorithms
Algorithm Gradient Transversal Fast K a l m a n Kalman Square root K a l m a n Gradient Lattice Least Square Lattice Generalized F T F
N u m b e r of multiplications
N u m b e r of divisions
2N + 1 20N + 5 2.5N2+4.5N 1.5N2+6.5N
0 3 2 N 2N 1 2N l 2
13NI+33N2-36 18Nt+39N2-39 12 N + 27
N denotes the n u m b e r of taps in the equalizer, where N = N~ + N 2 Signal Processing
In this paper, we have presented the generalized fast transversal filter algorithm which is applicable to the decision feedback equalization of dispersive channels. From Table 2, we observe that the generalized FTF algorithm requires less computational effort compared to the other fast algorithm. The mean squares error convergence rate of the algorithm and its comparison with the LMS algorithm provides additional motivation for the use of the generalized FTF algorithm. Also, a comparison of the steady state tracking/excess noise
247
D.K. Mehra / A generalized,fast transversal fiher algorithm
3o[
When differentiation with respect to an Ndimensional vector Z is required, we define the complex gradient operator Vz as
Channe[ (0.3,0.9,0.3) 6 = 0.001,7~ = 1.0 1.OIL..~
"~
~1\
A GFTF (7v2=0.001 ~
F/\
B GFTF ~V 2= 0,01
\
~ ~
[ 0 ( ' ) C)(') 0 ( ' ) 3 T,
<~:o.0ol
vz(.
): L
;-;J
' 0zN
where O(. )/azk, differentiation with respect to Zk, is carried out as defined earlier. Using the above, it may be shown that
Vz(A"Z)=A*,
(A.1)
Vz(ZHA) =0,
(A.2)
V z ( Z H R Z ) = (RZ)*.
(A.3)
It may be noted that the definition of the complex gradient operator used above is different from the definition of the complex gradient operator given in [4]. For a real scalar valued function of the complex variable, however, the results obtained in [4] may be obtained using the above definition
0.01 - -
I
O.OC
50
I
100 Number of iterations
I
150
175
3.0
•
Channet (0.L,08, 0.815, 0.408 )
Fig. 1. Convergence rate of GFTF and LMS algorithm for Channel 1.
8=0.001, ~=1.0
Appendix A. Complex gradient operator If z is a complex scalar quantity given by z = x +jy, then the following scalar complex derivatives are defined: a(z)_l 0z 2
0z_j
=1,
lrOz,
.oz*]
az =2Lax -J-'-
--'ayJ
a(z*)
a(z) az*
lraz
.az]
1 [az*
oez,) - 2
. +'
B GFTF 0~v2=0.01 C
LMS
O~'v2 = 0.001
D
LMS
O~-v2 = 0.01
I 0.1 3
g
0.0'
~0,
=o, 0.0
a(z*)
A GFTF O~v2=0.001
1.0
trade off between generalized FTF algorithm and stochastic gradient algorithm could prove useful for the application of the above algorithm to track time-varying channels.
az] = '
0
I
50
I
100 Number of iterations
I
150
175
-~-
Fig. 2. Convergence rate of GFTF and LMS algorithm for Channel 2. Vol.21, No. 3, November1990
248
D.K. Mehra / A generalized fast transversal filter algorithm
followed by a complex congugate operation of the result. Similarly, V w, the complex gradient operator with respect to the matrix W,
Differentiating (B.1) with respect to Cm(n) using (A.1)-(A.3) we get
(j=~oA "-J Xm(j)xH (j) ) C,,( n ) n
W~-
0)21
0)22
0)2M
0)NI
0)N2
0)NM
is defined as
Vw(" )=
+ • A n Jd*(j)Xm(j)=O,
,
j-O
which can be rewritten as
[ ._) 0(.) / i; 1),
0(.)
0tol2
R,,(n)C,,,(n)= ~/m(n),
0toiM
where
!
[.OWN~ OWNZ
n
OWN~
R,,,(n) = Z j-O
where 0)~k,the element of the matrix ~ is given by (Oik = Xik
(B.2)
A"-Jxm(j)X~(J)
(B.3)
and
+ jYik"
In the generalized FTF algorithm, the trace of the matrix function W is usually to be minimized. Typical matrix functions arising in the derivation of the above algorithm are of the form BHW, WHB and WHRW, where B is an N × M matrix. The gradients of the trace of these matrix functions with respect to the components of W are given by the following equations; V w(trace(B HW)) = B*,
(A.4)
V w(trace(WHB)) = 0,
(A.5)
V w(trace(WHRW)) = (RW)*.
(A.6)
~%(n) = ~ A"-Jd*(j)Xm(j).
(B.4)
j=o
Using ( l l b ) in (12) and differentiating the trace of the resulting equation with respect to Amp(n), and using (A.4)-(A.6), we get
Rm(n-1)amp(n)= -~'mp(n), f
(B.5)
where y~p(n) = ~ A" JX,,(j--1)eH(j).
(B.6)
j=o
Using (B.3), Rm(n) can be updated as
Rm(n + 1) = ARm(n)+Xm(n+ 1)XH(n+ 1). Appendix B. Summary of the derivation of the generalized FTF algorithm
Similarly, from (B.4) ~m(n + 1) can be written as •/,,(n + 1) = A~m(n) + d*(n + 1)Xm(n + 1). (B.8)
Using (3), (2) can be written as n
E x"-J[d(J) + C~(n)Xm(j)]
Using (B.7), (B.8) and (B.2), we can write
j=O
AR,,,(n)C,,,(n+l)
× [d(j)+ C~(n)Xm(j)]*
= AR,,,(n)C,,(n) - X,,(n + 1)
= Z a"-S[d(j)d*(j) j-O
x [d(n+ 1)+ C~(n + 1)Xm(n+ 1)]*, + CH(n)Xm(j)d*(j)
which may be rewritten as
+ [d*(j)Xm(j)]HCm(n) + CH(n)X,,(j)X~(j)Cm(n)]. Signal Processing
(B.7)
C,,,(n + 1) = Cm(n) + W,,(n + 1)e*m(n + 1), (B.1)
(B.9)
D.K. Mehra / A generalizedfast transversalfilter algorithm
249
Using (B.12) and (7) in (B.16), we have
where
Wm(n + 1) =
1
-~- R~'(n)X,,,(n
+ 1).
(B.10)
F Using (B.10) in (B.5), (B.7) and (B.8), we can prove (19). Using (7), we can show that the correlation matrix /~m(n), of the extended vector X,,(n) can be updated as
~SM
T
_(1/A)[Ef,,,(n)] 1
ef,.(n + l ) ]
! ..,
W.,(n)-(1/h)A,,p(n)[Ef,(n)]
'
ef,(n+l)J (B.17)
Similarly, using (B.13) and (9) in (B.15),
RM(n+ I)= ARM(n) + XM(n + 1)[.~M (n + 1)] H.
(B.11)
QMIYCM(n+I) eb.,(n+l)]
Using the matrix inversion lemma and (8), ( l l ) and (B.7), we can show that
[SMRM(n)sT]-l=[; R~,'(n0 1)1
k
-(1/h)[E~,(n)]'
e,~,In+l)J (B.18)
I H +[Amf(n)] [El(n)] ' [A,.p(n)] " (B'12)
The gain constants a,,(n + 1) and aM(n + 1) are, respectively, defined as
am(n+ 1) = 1 - [ W m ( n + 1)]HXm(n+ 1), (B.19)
Similarly, we can write
aM(n + 1) = 1 --[ l$'M(n + 1)]HXM(n + 1) (B.20) or
aM(n+ 1) = 1--[QMWM(n+ 1)] H Using the following definition for the extended gain vector, IVM(n + 1):
× QMXM(n + 1).
(B,21)
Using (B.18), (9) in (B.21), we get ff'M(n+l)=-l[/~M(n)]
'2M(n+l),
aM(n + 1) = am(n+ 1 ) + 1 [e~(n + 1)] H
(B.14)
A
x [ E b ( n ) ] 'eb(n + 1).
we can rewrite
ff'M(n+l)
(B.22)
Similarly, (B.22) can also be written using (B.17) and (8) as
1 QVM[QM~M(n)QT]_,QM~M( n + 1)
h
aM(n + 1) = a,,(n)+ 1 (e~(n + 1))" A
(B.15)
x[E~(n)]-'e~(n+l).
or
W~(n+l)
(B.23)
It may be shown easily that (21) can also be written as
1
T
--
T
A SMRM(n)S~]
--1
--
S~4XM(n+I).
C,,,(n + 1) = C,,,(n)+ W ' ( n + 1)e*(n + 1), (B.16)
(B.24) Vol. 21, No. 3, November 1990
D.K. Mehra / A generalized jast transversal.filter algorithm
250
where
we get
W'(n+
1 1) = --7A
R~'(n + 1)X~(n
em(n + 1)
+ 1).
(B.25)
em(n+l) a,.(n + 1)"
(B.28)
In a similar manner, we can prove (31) and (32).
Using (B.7) and the matrix inversion lemma, References
R,,,J(n+ l)=lR~l(n) --1
A
R~(n)X,,(n+l)l X mH( n + l ) R ~ l ( n ) A 1
1 + A X H ( n + 1 ) R m l ( n ) X m ( n + 1)
R~'(n+ l)=lR.,'(n) A
W m ( n + l) W H ( n + l) ce,,(n + 1)
(B.26) Post-multiplying (B.26) by
X,n(n + 1),
W',(n+l)=Wm(n+l)
Wm(n+ l)[1-ol,,,(n+ l)] .,.(n+l) w,.(n+ 1) W'(n + 1 ) - am(n + 1)"
(B.27)
Using (B.27) in (B.24) and comparing with (21),
Signal Processing
[1] J.M. Cioffi and T. Kailath, "Fast recursive least squares transversal filters for adaptive filtering", IEEE. Trans. Acoust. Speech Signal Process., Vol. 32, April 1984, pp. 304-337. [2] D.D. Falconer and L.Ljung, "Application of fast Kalman estimation to adaptive equalization", 1EEE Trans. Comm., Vol. COM-26, October 1978, pp. 1439-1446. [3] D. Godard, "Channel equalization using a Kalman filter for fast data transmission", IBM J. Res. Develop., May 1974, pp. 267-273. [4] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, N J, 1986. [5] F.M. Hsu, "Square-root Kalman filtering for high speed data received over fading dispersive HF channels", IEEE Trans. Inform. Theory, Vol. IT-28, September 1982, pp. 753-763. [6] F. Ling and J.G. Proakis, "A generalized multichanneJ least squares lattice algorithm with sequential processing stages", IEEE Trans. Acoust. Speech Signal Process., Vol. 32, April 1984, pp. 381-389. [7] F. Ling and J.G. Proakis, "Adaptive lattice decision feedback equalizers--Their performance and application to time-variant multipath channels", IEEE Trans. Comm., Vol. COM-33, April 1985, pp. 348-356. [8] J.G. Proakis, Digital Communications, McGraw-Hill, New York, 1983. [9] B. Widrow and Hoff, "Stationary and nonstationary learning characteristics of the LMS adaptive filter", Proc. IEEE, Vol. 64, August 1976, pp. 1156-1162.
241
A G E N E R A L I Z E D LEAST S Q U A R E S FAST TRANSVERSAL FILTER A L G O R I T H M FOR THE D E C I S I O N FEEDBACK E Q U A L I Z A T I O N OF DISPERSIVE C H A N N E L S D.K. M E H R A Department of Electronics and Computer Engineering, University of Roorkee, Roorkee, India 247-667 Received 29 February 1988 Revised 27 December 1988 and 25 May 1989
Abstract. Recently computationally efficient recursive least squares adaptive filtering algorithms have been introduced in transversal filter form as well as in lattice form. Fast transversal filter implementation of the generalized recursive least squares adaptive algorithm, which is appropriate for multichannel adaptive filtering and estimation, is presented in this paper. The above algorithm is derived in particular for the decision feedback equalization of dispersive channels, when the number of stages in the forward and backward filters are different. The algorithm considered in this paper is advantageous over the generalized least squares lattice algorithm considered by Ling and Proakis, because of the less computational effort required. The convergence rate of~the mean square error for the decision feedback equalization of QPSK data transmitted through the dispersive channel using the generalized fast transversal filter algorithm is obtained using computer simulations. Zusammenfassung. Kfirzlich sind recheneffiziente rekursive adaptive Filteralgorithmen auf der Basis Kleinster Quadrate sowohl in Transversal als auch in lattice-Form eingeffihrt worden. In diesem Beitrag wird eine schnelle Realisierung als Transversalfilter ffir den verallgemeinerten rekursiven adaptiven Kleinste-Quadrate-Algorithmus vorgestellt, der ffir die mehrkanalige adaptive Filterung und Sch~itzung geeignet ist. Der genannte Algorithmus wird insbesondere hergeleitet ffir die Entzerrung dispersiver Kanfile mit Entscheidungs-Rfickkopplung ffir den Fall, da D die Anzahlen der Stufen in den Vorwiirts- und Rfickwartsfiltern unterschiedlich sind. Der hier behandelte Algorithmus weist gegenfiber dem entsprechenden Latticealgorithmus, wie ihn Ling und Proakis betrachten, Vorteile auf, da er weniger Rechenaufwand erfordert. Aus Rechnersimulationen werden Angaben fiber die Konvergenzgeschwindigkeit des mittleren quadratischen Fehlers des schnellen verallgemeinerten Transversalfilteralgorithmus bei einer QSPK-Daten-Entzerrung mit Entscheidungsrfickkopplung und nach der l]bertragung fiber einen dispersiven kanal gewonnen.
R6sum6. R6cemment des algorithmes r6cursifs de filtrage adaptatif aux moindres carr6s efficaces au niveau calcul ont 6t6 introduits aussi bien sous la forme transversale qu'en treillis. L'implantation sur un filtre transversal rapide de l'algorithme r6cursif adaptatif aux moindres carr6s g6n6ralis6s, qui se pr&e bien aux op6rations de filtrage adaptatif multicanal et d'estimation est pr6sent6 dans cet article. Cet algorithme est sp6cifi6 en particulier pour l'egalisation par d6cision en mode r6trograde de cannaux dispersifs, quand les tailles des filtres avant et arri~re sont diff6rentes. L'algorithme consid6r6 dans cet article est plus avantageux que l'algorithme aux moindres carr6s g6n6ralis6s en treillis pr6sent6 par Ling et Proakis, car il n6cessite moins de calculs. Le taux de convergence de l'erreur quadratique moyenne lors de 1'6galisation en mode de d6cision r6trograde de donn6es QPSK transmises fi travers un canal dispersif ~ I'aide de I'algorithme de filtre transversal rapide g6n6ralis6 est obtenue a l'aide de simulations sur ordinateur.
Keywords. Recursive estimation, generalized fast transversal filtering, decision feedback equalization.
1. Introduction In high speed
digital communication,
use of the available
channel
bandwidth
limited by the presence of the intersymbol
efficient is o f t e n interfer-
0165-1684/90/$03.50 © 1990 - Elsevier Science Publishers B.V.
ence caused by the dispersion of the channel. The linear equalizer [8] widely used for the equalization of such channels employs a transversal filter structure, whose tap coefficients are adjusted adaptively using the gradient type LMS algorithm of
242
D.K. Mehra / A generalized.fast transversal filter algorithm
Widrow and Hoff [9]. On channels that have spectral nulls, the linear equalizer yields poor error rate performance. In such situations, the decision feedback equalizer which has a very similar computational complexity, but a better error rate performance, is used. The simplest adaptive DFE consists of two transversal filters, a feedforward filter and a feedback filter, and uses the WidrowHoff type LMS algorithm to adapt its coefficients [9]. Although the computational complexity of these algorithms is of the order of 2 N to 3N, where N is the number of equalizer tap coefficients, these algorithms are slower to converge initially. Initial convergence of the algorithm based on the Kalman algorithm [3] is very fast, but a disadvantage of the Kalman algorithm is its computational complexity, where N 2 operations are to be performed per iteration. Recently computationally efficient recursive least square (RLS) adaptive algorithms have been introduced. These algorithms require a number of arithmetic operations per iteration that is proportional to the number of variable parameters in the adaptive filter. These algorithms simultaneously offer the improved convergence properties over LMS algorithms. RLS algorithms have been introduced in transversal filter form as well as in lattice filter form. Recently least squares lattice DFE and gradient lattice DFE algorithms have been developed and their application to the equalization of fixed and time varying multipath channels have also been considered [6, 7]. Lattice algorithms implement RLS algorithms by simultaneous order and time updating. In equalizer applications, lower order solutions are generally not essential, a reduction in computational r'equirement in comparison to the lattice form is achieved through the use of transversal filter algorithms. Fast transversal filter algorithms (FTF) introduced in [1] are the fastest known RLS algorithms and when considered for linear equalization, their computational complexity is proportional to 7N. In this paper, we have developed an FTF DFE algorithm, which differs from the conventional multichannel FTF algorithm [ 1] which is restricted Signa~ Processing
to have the same number of stages in the feedforward and the feedback stages. The FTF algorithm derived is applicable in a general situation, when the number of stages in forward and backward filters are different. We also consider the application of the above algorithm to the DFE of fixed dispersive channel, when QPSK data is transmitted over such channels. Convergence rate of the mean squares error for the above situation is determined using computer simulations. The organisation of this paper is as follows. The generalized least squares filtering problem is introduced in Section 2 and the generalized FTF algorithm is derived in Section 3. The computational complexity of the algorithm when applied to the decision feedback equalization of dispersive channels and its comparison to some other DFE algorithms is given in Section 4. In Section 5 we finally consider the application of the above algorithm and also give some simulation results.
2. Generalized least squares filtering The classical least squares problem is to find the adaptive filter coefficients at each time instant so as to minimize the accumulation of squared error between the filter output and the desired output up to that time instant. For example in an equalizer adjustment algorithm, suppose u(0), u ( 1 ) , . . . , u(n) is a training sequence of complex data symbols transmitted over a channel represented by a tapped delay line filter structure with tap spacing equal to the symbol interval. Channel introduces intersymbol interference and additive noise. Further assume that the sequence {u(n)} transmitted over a channel results in the received sequence as {y(n)}. To retrieve the transmitted sequence, the DFE uses the linear combination of the received signal y(n) and its past values y(n - i), i = 1 , 2 , . . . , N~ - 1. In addition, it uses the information about previously decoded values d ( n - 1), d(n - 2 ) , . . . , d(n - N2) of the transmitted sequence.
D.K. Mehra / A generalizedfast transversalfilter algorithm We thus define the following m-dimensional vector Xm(n), consisting of time delayed samples of y(n) and the past decision as
243
where R,,,(n) is an m x m correlation matrix given by
R,n(n)= ~ A n iXm(j)X~(j) , i
y(n)
(5)
0
and the cross correlation vector ~,,(n) is given by
y(n-1)
n
X~(n) = v ( n - N , + I d(n-1) d(n - 2 )
,
(1)
(6)
X.,(n) defined in (1) is an (N~+ N2)-dimensional vector and, similar to the ease of
The data vector
d(n - N2) where
m=NI+N2. The least squares problem in DFE is then to find the m-dimensional vector Cm(n) of equalizer tap values, which minimizes the following sum of squared errors:
A" i e,,(j) e*(j),
~ , , ( n ) = ~" An i[d*(j)Xm(j)].
(2)
the linear equalizer, a certain shifting property exists for X,.(n). Without loss of generality we assume that N~ ~> N2, and using the permutation matrices SM and QM defined in [12] for deriving the fast Kalman algorithm, we ean show, using the definition of the extended vector
~'m(n + 1) = [y(n + 1 ) , y ( n ) , . . . , y ( n -- N, + 1), d ( n ) , . . , d ( n - N:)] x,
(7)
that
.j-o
SMRM(n + 1) = L X.,(n where the exponential forgetting factor A is such that 0~
(3)
and * denotes the complex conjugate operation. In (3)
where M = m + 2 , p = 2 and ep(n + l) is the vector consisting o f p new elements entering the equalizer given by
Similarly
-Xm(n+l)] QMXM(n + 1) . . . . ,
(9)
p~(n + l) J
C,,(n) = [ c , ( n ) , . . . , CN,(n), CN,+,(n),
where pp(n + 1) is the vector consisting o f p deleted old elements given by
. . . . cN,+N~(n)] ~, where c~(n), i= 1 , 2 , . . . , Nj, are the tap gains of the forward filter, and cN,+~(n), i = 1, 2 , . . . , N2, are the tap gains of the backward filter. Using the definition of the complex gradient operator given in Appendix A, we can show that the optimum value of the vector Cm(n) which minimizes the value of (2) is given by
C,,(n) = -Rm'(n)~m(n),
(8)
[v(n+l)] e p ( n + l ) = [ _ " d(n) J"
the a posteriori error residual, e,,(n), is
era(n) = d(n)+ C~(n)Xm(n)
'
(4)
[Y(n-N,+I) 1 d ( n - N 2 ) 3"
Pr(n+l)=[_
3. Generalized FFF algorithm
It is convenient to carry out the derivation of the algorithm in terms of the a priori error residual Vol 21. No 3, No~ember 1990
244
D.K. Mehra / A generalizedfast transversalfilter algorithm
e,. (n) defined as
e,.(n)=d(n)+C~(n-1)X.,(n).
(10)
In addition, a priori and a posteriori forward prediction error residual vectors are defined as
er~(n)=ep(n)+A~p(n-1)Xm(n-1),
(lla)
e~(n)=er(n)+a~p(n)X~.(n-1),
(llb)
where the m xp tap gain matrix Amp(n) of the foreword prediction error filter is recursively adapted to minimize the trace of
It is shown in Appendix B that the tap gains of the forward, backward and equalizer filter can be recursively updated as
A.,p(n+l)=a,.p(n)+ Wm(n)[e~(n+l)] H, Bmp(n + 1) = Br.p(n)+ Wm(n + 1)[eb(n + 1)] H,
(20) C,.(n+l)=Cm(n)+Wm(n+l)e*(n+l),
j=o
n-j
f
•
f
•
H
em(J)(em(J) ) .
Rm(n-1)Amp(n)= -Vmp(n), r
Wm(n + 1) = - 1 R ~ ' ( n ) X m ( n + l ) .
(12)
It is shown in Appendix B that the optimum value of Amp(n) is given by
(22)
Similarly we may define the extended gain vector as 1
(13)
where the cross-correlation matrix 7~p(n) is given by
(21)
where the gain vector Wm(n + 1) of the algorithm is defined as
n
2 A
(19)
l~'M(n + 1) = --~-/~ml(n).~M(n + 1),
(23)
where/~M (n) is the correlation matrix of the extended vector -~M(n + 1) given by
n
V~p(n)= ~ 3." Jxm(j--1)eHp(j).
(14)
j = 0
It can easily be shown that the minimum value of (12) can be recursively updated as
E~(n + 1) = AE~(n) + eCm(n+ 1)[e~(.+,)] H. (15)
/~M(n+l)= ~ An-J[f(M(n+I)][XM(n+I)]H. j-o In Appendix B, it is shown that the extended gain vector 1~,"M(n + 1) can be recursively updated as ~.(n+l)
_(I/A)[Ef.,(n)]_ '
The a priori and a posteriori backward error residuals are defined as
eb(n)=pp(n)+B~p(n-l)Xm(n),
(16a)
eb(n) = pp(n) + B~p(n)X,.(n),
(16b)
where the m x p tap gain matrix Bmp(n ) of the backward prediction error filter is recursively updated to minimize the trace of A n--Jeb (j)[ eb.n(j) ]H.
= ST~
.
.
.
er.,(n + 1)] .
.
.
(24)
or QMWM(n+l)
[
Wm(n+l)-(1/A)B,,w(n)[E~(n)]-'
(17)
...
Eb(n + 1) = AEb.,(n) + eb(n + 1)[eb(n + 1)] H. (18) Signal
Processing
eb(n+ 1)] ...
- (1/A)[E~(n)]-t
.
eb(n + 1)
j--0 It may be shown easily that the minimum value of (17) can be recursively updated as
.
W,,,(n)_(1/A)Amp(n)[Ef(n)] t er,,,(n+l)
(25) Define
QM[ffCM(n+ I)] . . . .
.
(26)
D.K. Mehra / A generalized fast transversal filter algorithm
5. Simulation results
From (25) and (26) we get W~,(n + 1) = ff'm(n+l)-Bmp(n)l.%(n+l), (27)
eb(n + 1) = -AEb(n)/xp(n + 1).
(28)
Further, the scalar constant am(n) is defined as
am(n) = 1 - X~(n) W,~(n)
245
The generalized FTF algorithm considered above is used for the adaptive equalization of two linear dispersive communication channels, whose discrete time equivalent channel impulse responses are
(29)
Channel 1
(0.30, 0.90, 0.30),
and it is shown in Appendix B that ~,.(n) can be recursively updated as
Channel 2
(0.408, 0.815, 0.408).
ct,,(n+ l)=ctm(n) + 1 [e f ( n + 1)]H[Ef (n)]_~ef ( n+ 1) + [eb~(n + 1)]"tZp(n + 1).
(30)
The a posteriori error residual vectors can be updated using the a priori error residual vector and scalar constant a~(n) as
e~(n + 1) - er''(n + 1) ~m(n)
e b ( n +1)
(31) '
eb(n + 1) a.,(n + 1)'
Channel 1 has eigenvalue ratio of 25, while Channel 2 has eigen value ratio of a. The input to the channel is a uniformly distributed complex sequence, where each element of the sequence takes any one of the four values from the set (l+j,-l+j,-1-j,l-j) thus representing the QPSK data. For the two channels under consideration, complex additive Gaussian noise with variance 2 =0.001 or 0.01 corrupts the signal. The decision feedback equalizer is assumed to have 9 forward taps and 2 backward taps. Initial conditions for the generalized FTF algorithm are
(32) 0
e . , ( n + 1)
em(n+ l)=ol,.(n+ l ).
(33)
Wm(O ) = Cm(O )
=Omxl,
A.,p(O) = Brae(O) = Ore×p, 4. Generalized FFF algorithm and its computational complexity We can rewrite the generalized FTF algorithm using the scalar constant %,(n) defined as 1 w(n)
=
~(n)"
The algorithm is listed in Table 1, along with the computational complexity of each step, when A = 1. The computational complexity has been computed in terms of required number of multiplications. The complexity of some of the adaptive DFE algorithms is listed in Table 2.
where 0 is a null matrix of dimensions as indicated. The mean squares error (MSE) convergence rate performance for the generalized FTF and LMS algorithms is obtained by averaging the MSE over 100 individual learning curves. The LMS algorithm uses a constant step size ~ = 0.025. Figures 1 and 2 show the convergence rate of the generalized FTF algorithm with A =1 and of the LMS algorithm for Channels 1 and 2, respectively. It may be seen from the figures that for the G F T F algorithm, the convergence rate is independent of the channel eigenvalue ratio and the G F T F algorithm converges significantly faster than the LMS algorithm. Vol. 21, No. 3, November |990
246
D.K. Mehra
A generalized fast transversal fiher algorithm
Table 1
Generalized FTF algorithm and c o m p u t a t i o n a l complexity Computation
Complexity
ef,,,(n+ 1) = el,(n+ 1)+ A~w(n)X,,,(n)
mp
Ee,.(n + 1) = e~,,(n + 1) y,,,(n)
P
-(lla)[U,.(n)]' l ~ a (n +
l)
= sT
ef'"(n+l)]...
mp + p2
...
W.,(n)-(lt.t)A,..(n)[Ef.,(n)]
'
e~,,(n + 1)
A,,,~,(n+ 1) = A,,,p(n) + W,,,(n)[e~,,(n+ 1)] H
mp
Ef.,(n + 1) = AEf.,(n)+ e~,,(n + 1)[e~,,(n + 1)] H
p:
[ #,°(n+,)] O~[#M(n + 1)] .
. . .
/z¢,(n + 1) _] W,,,(n + 1) = l,V,,,(n+l)-B.,p(n)~p(n+l)
mp p2
e~,(n + 1 ) = -AE~,(n)~.(n + 1) v,,,(n)
~M (n + 1) - 1 + (1/A)[ ef.,(n + 1)]H[ Elm(n)] - ' er,,,(n + 1) y,.(n + 1)
yM(n+ l) 1 + [eb,(n + l)]H/z/,(n + 1)VM (n + 1)
eb,,,(n + 1) = e~,(n + 1)ym(n + 1)
P 2p
E~,(n + 1) = A E,b,,(n ) + e,b,,(n + 1)[e~,( n + 1)]H
P pZ
B,,,(n + l) = B.,(n)+ W,,,(n + l)[~b,(n + l)] H
mp
e,,,(n + l) = d(n + 1)+ C~,(n)X,.,(n + I)
m
e,.(n+ l ) = e , . ( n + l)y.,(n+ l)
1
C,,(n
+ 1) =
C,,,(n)+ W,,,(n + l)e*,(n + 1)
m
6. Conclusion
Table 2 C o m p u t a t i o n a l complexity of adaptive D F E algorithms
Algorithm Gradient Transversal Fast K a l m a n Kalman Square root K a l m a n Gradient Lattice Least Square Lattice Generalized F T F
N u m b e r of multiplications
N u m b e r of divisions
2N + 1 20N + 5 2.5N2+4.5N 1.5N2+6.5N
0 3 2 N 2N 1 2N l 2
13NI+33N2-36 18Nt+39N2-39 12 N + 27
N denotes the n u m b e r of taps in the equalizer, where N = N~ + N 2 Signal Processing
In this paper, we have presented the generalized fast transversal filter algorithm which is applicable to the decision feedback equalization of dispersive channels. From Table 2, we observe that the generalized FTF algorithm requires less computational effort compared to the other fast algorithm. The mean squares error convergence rate of the algorithm and its comparison with the LMS algorithm provides additional motivation for the use of the generalized FTF algorithm. Also, a comparison of the steady state tracking/excess noise
247
D.K. Mehra / A generalized,fast transversal fiher algorithm
3o[
When differentiation with respect to an Ndimensional vector Z is required, we define the complex gradient operator Vz as
Channe[ (0.3,0.9,0.3) 6 = 0.001,7~ = 1.0 1.OIL..~
"~
~1\
A GFTF (7v2=0.001 ~
F/\
B GFTF ~V 2= 0,01
\
~ ~
[ 0 ( ' ) C)(') 0 ( ' ) 3 T,
<~:o.0ol
vz(.
): L
;-;J
' 0zN
where O(. )/azk, differentiation with respect to Zk, is carried out as defined earlier. Using the above, it may be shown that
Vz(A"Z)=A*,
(A.1)
Vz(ZHA) =0,
(A.2)
V z ( Z H R Z ) = (RZ)*.
(A.3)
It may be noted that the definition of the complex gradient operator used above is different from the definition of the complex gradient operator given in [4]. For a real scalar valued function of the complex variable, however, the results obtained in [4] may be obtained using the above definition
0.01 - -
I
O.OC
50
I
100 Number of iterations
I
150
175
3.0
•
Channet (0.L,08, 0.815, 0.408 )
Fig. 1. Convergence rate of GFTF and LMS algorithm for Channel 1.
8=0.001, ~=1.0
Appendix A. Complex gradient operator If z is a complex scalar quantity given by z = x +jy, then the following scalar complex derivatives are defined: a(z)_l 0z 2
0z_j
=1,
lrOz,
.oz*]
az =2Lax -J-'-
--'ayJ
a(z*)
a(z) az*
lraz
.az]
1 [az*
oez,) - 2
. +'
B GFTF 0~v2=0.01 C
LMS
O~'v2 = 0.001
D
LMS
O~-v2 = 0.01
I 0.1 3
g
0.0'
~0,
=o, 0.0
a(z*)
A GFTF O~v2=0.001
1.0
trade off between generalized FTF algorithm and stochastic gradient algorithm could prove useful for the application of the above algorithm to track time-varying channels.
az] = '
0
I
50
I
100 Number of iterations
I
150
175
-~-
Fig. 2. Convergence rate of GFTF and LMS algorithm for Channel 2. Vol.21, No. 3, November1990
248
D.K. Mehra / A generalized fast transversal filter algorithm
followed by a complex congugate operation of the result. Similarly, V w, the complex gradient operator with respect to the matrix W,
Differentiating (B.1) with respect to Cm(n) using (A.1)-(A.3) we get
(j=~oA "-J Xm(j)xH (j) ) C,,( n ) n
W~-
0)21
0)22
0)2M
0)NI
0)N2
0)NM
is defined as
Vw(" )=
+ • A n Jd*(j)Xm(j)=O,
,
j-O
which can be rewritten as
[ ._) 0(.) / i; 1),
0(.)
0tol2
R,,(n)C,,,(n)= ~/m(n),
0toiM
where
!
[.OWN~ OWNZ
n
OWN~
R,,,(n) = Z j-O
where 0)~k,the element of the matrix ~ is given by (Oik = Xik
(B.2)
A"-Jxm(j)X~(J)
(B.3)
and
+ jYik"
In the generalized FTF algorithm, the trace of the matrix function W is usually to be minimized. Typical matrix functions arising in the derivation of the above algorithm are of the form BHW, WHB and WHRW, where B is an N × M matrix. The gradients of the trace of these matrix functions with respect to the components of W are given by the following equations; V w(trace(B HW)) = B*,
(A.4)
V w(trace(WHB)) = 0,
(A.5)
V w(trace(WHRW)) = (RW)*.
(A.6)
~%(n) = ~ A"-Jd*(j)Xm(j).
(B.4)
j=o
Using ( l l b ) in (12) and differentiating the trace of the resulting equation with respect to Amp(n), and using (A.4)-(A.6), we get
Rm(n-1)amp(n)= -~'mp(n), f
(B.5)
where y~p(n) = ~ A" JX,,(j--1)eH(j).
(B.6)
j=o
Using (B.3), Rm(n) can be updated as
Rm(n + 1) = ARm(n)+Xm(n+ 1)XH(n+ 1). Appendix B. Summary of the derivation of the generalized FTF algorithm
Similarly, from (B.4) ~m(n + 1) can be written as •/,,(n + 1) = A~m(n) + d*(n + 1)Xm(n + 1). (B.8)
Using (3), (2) can be written as n
E x"-J[d(J) + C~(n)Xm(j)]
Using (B.7), (B.8) and (B.2), we can write
j=O
AR,,,(n)C,,,(n+l)
× [d(j)+ C~(n)Xm(j)]*
= AR,,,(n)C,,(n) - X,,(n + 1)
= Z a"-S[d(j)d*(j) j-O
x [d(n+ 1)+ C~(n + 1)Xm(n+ 1)]*, + CH(n)Xm(j)d*(j)
which may be rewritten as
+ [d*(j)Xm(j)]HCm(n) + CH(n)X,,(j)X~(j)Cm(n)]. Signal Processing
(B.7)
C,,,(n + 1) = Cm(n) + W,,(n + 1)e*m(n + 1), (B.1)
(B.9)
D.K. Mehra / A generalizedfast transversalfilter algorithm
249
Using (B.12) and (7) in (B.16), we have
where
Wm(n + 1) =
1
-~- R~'(n)X,,,(n
+ 1).
(B.10)
F Using (B.10) in (B.5), (B.7) and (B.8), we can prove (19). Using (7), we can show that the correlation matrix /~m(n), of the extended vector X,,(n) can be updated as
~SM
T
_(1/A)[Ef,,,(n)] 1
ef,.(n + l ) ]
! ..,
W.,(n)-(1/h)A,,p(n)[Ef,(n)]
'
ef,(n+l)J (B.17)
Similarly, using (B.13) and (9) in (B.15),
RM(n+ I)= ARM(n) + XM(n + 1)[.~M (n + 1)] H.
(B.11)
QMIYCM(n+I) eb.,(n+l)]
Using the matrix inversion lemma and (8), ( l l ) and (B.7), we can show that
[SMRM(n)sT]-l=[; R~,'(n0 1)1
k
-(1/h)[E~,(n)]'
e,~,In+l)J (B.18)
I H +[Amf(n)] [El(n)] ' [A,.p(n)] " (B'12)
The gain constants a,,(n + 1) and aM(n + 1) are, respectively, defined as
am(n+ 1) = 1 - [ W m ( n + 1)]HXm(n+ 1), (B.19)
Similarly, we can write
aM(n + 1) = 1 --[ l$'M(n + 1)]HXM(n + 1) (B.20) or
aM(n+ 1) = 1--[QMWM(n+ 1)] H Using the following definition for the extended gain vector, IVM(n + 1):
× QMXM(n + 1).
(B,21)
Using (B.18), (9) in (B.21), we get ff'M(n+l)=-l[/~M(n)]
'2M(n+l),
aM(n + 1) = am(n+ 1 ) + 1 [e~(n + 1)] H
(B.14)
A
x [ E b ( n ) ] 'eb(n + 1).
we can rewrite
ff'M(n+l)
(B.22)
Similarly, (B.22) can also be written using (B.17) and (8) as
1 QVM[QM~M(n)QT]_,QM~M( n + 1)
h
aM(n + 1) = a,,(n)+ 1 (e~(n + 1))" A
(B.15)
x[E~(n)]-'e~(n+l).
or
W~(n+l)
(B.23)
It may be shown easily that (21) can also be written as
1
T
--
T
A SMRM(n)S~]
--1
--
S~4XM(n+I).
C,,,(n + 1) = C,,,(n)+ W ' ( n + 1)e*(n + 1), (B.16)
(B.24) Vol. 21, No. 3, November 1990
D.K. Mehra / A generalized jast transversal.filter algorithm
250
where
we get
W'(n+
1 1) = --7A
R~'(n + 1)X~(n
em(n + 1)
+ 1).
(B.25)
em(n+l) a,.(n + 1)"
(B.28)
In a similar manner, we can prove (31) and (32).
Using (B.7) and the matrix inversion lemma, References
R,,,J(n+ l)=lR~l(n) --1
A
R~(n)X,,(n+l)l X mH( n + l ) R ~ l ( n ) A 1
1 + A X H ( n + 1 ) R m l ( n ) X m ( n + 1)
R~'(n+ l)=lR.,'(n) A
W m ( n + l) W H ( n + l) ce,,(n + 1)
(B.26) Post-multiplying (B.26) by
X,n(n + 1),
W',(n+l)=Wm(n+l)
Wm(n+ l)[1-ol,,,(n+ l)] .,.(n+l) w,.(n+ 1) W'(n + 1 ) - am(n + 1)"
(B.27)
Using (B.27) in (B.24) and comparing with (21),
Signal Processing
[1] J.M. Cioffi and T. Kailath, "Fast recursive least squares transversal filters for adaptive filtering", IEEE. Trans. Acoust. Speech Signal Process., Vol. 32, April 1984, pp. 304-337. [2] D.D. Falconer and L.Ljung, "Application of fast Kalman estimation to adaptive equalization", 1EEE Trans. Comm., Vol. COM-26, October 1978, pp. 1439-1446. [3] D. Godard, "Channel equalization using a Kalman filter for fast data transmission", IBM J. Res. Develop., May 1974, pp. 267-273. [4] S. Haykin, Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, N J, 1986. [5] F.M. Hsu, "Square-root Kalman filtering for high speed data received over fading dispersive HF channels", IEEE Trans. Inform. Theory, Vol. IT-28, September 1982, pp. 753-763. [6] F. Ling and J.G. Proakis, "A generalized multichanneJ least squares lattice algorithm with sequential processing stages", IEEE Trans. Acoust. Speech Signal Process., Vol. 32, April 1984, pp. 381-389. [7] F. Ling and J.G. Proakis, "Adaptive lattice decision feedback equalizers--Their performance and application to time-variant multipath channels", IEEE Trans. Comm., Vol. COM-33, April 1985, pp. 348-356. [8] J.G. Proakis, Digital Communications, McGraw-Hill, New York, 1983. [9] B. Widrow and Hoff, "Stationary and nonstationary learning characteristics of the LMS adaptive filter", Proc. IEEE, Vol. 64, August 1976, pp. 1156-1162.